classification of functions we may classify functions by their formula as follows: polynomials...
TRANSCRIPT
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Classification of Functions
We may classify functions by their formula as follows:• Polynomials Linear Functions, Quadratic Functions. Cubic
Functions.• Piecewise Defined Functions Absolute Value Functions, Step Functions• Rational Functions• Algebraic Functions• Trigonometric and Inverse trigonometric
Functions• Exponential Functions• Logarithmic Functions
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Function’s Properties We may classify functions by some of their
properties as follows:• Injective (One to One) Functions• Surjective (Onto) Functions • Odd or Even Functions• Periodic Functions• Increasing and Decreasing
Functions• Continuous Functions• Differentiable Functions
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Power Functions
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Combinations of Functions
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Composition of Functions
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Inverse Functions
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Exponential Functions
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Logarithmic Functions
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The logarithm with base e is called the natural logarithm and has a special notation:
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Correspondence between degree and radian
The Trigonometric Functions
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Some values of and
sin cos
Trigonometric Identities
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Graphs of the Trigonometric Functions
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When we try to find the inverse trigonometric functions, we have a slight difficulty. Because the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that hey become one-to-one.
Inverse Trigonometric Functions
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The Limit of a Function
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Calculating Limits Using the Limit Laws
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Infinite Limits; Vertical Asymptotes
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Limits at Infinity; Horizontal Asymptotes
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Tangents• The word tangent is derived from the Latin word tangens, which
means “touching.”• Thus, a tangent to a curve is a line that touches the curve. In
other words, a tangent line should have the same direction as the curve at the point of contact. How can his idea be made precise?
• For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once. For more complicated curves this definition is inadequate.
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Instantaneous Velocity; Average Velocity• If you watch the speedometer of a car as you travel in city
traffic, you see that the needle doesn’t stay still for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined?
• In general, suppose an object moves along a straight line according to an equation of motion , where is the displacement (directed distance) of the object from the origin at time . The function that describes the motion is called the position function of the object. In the time interval from to
the change in position is . The average velocity over this time interval is
)(tfs s
t f
at hat )()( afhaf
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• Now suppose we compute the average velocities over shorter and shorter time intervals . In other words, we let approach . We define the velocity or instantaneous velocity at time to be the limit of these average velocities:
• This means that the velocity at time is equal to the slope of the tangent line at .
],[ haa h0
at
at P
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The Derivative of a Function
1
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Differentiable Functions
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The Derivative as a Function
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What Does the First Derivative Function Say about the Original Function?
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What Does the Second Derivative Function Say about the Original Function?
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Indeterminate Forms and L’Hospital’s Rule
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Antiderivatives
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